I have to find the first six terms of the sequence for the Mandelbrot set for c = 1i/2 and c=2. I was wondering, how do I go about this?
To find the first six terms of the sequence for the Mandelbrot set for the given values of c, which are 1i/2 and 2, you need to follow these steps:
1. Start with the first term, which is always c itself.
- For c = 1i/2, the first term is 1i/2.
- For c = 2, the first term is 2.
2. Compute the next term by squaring the previous term and adding c.
- For c = 1i/2, the second term is (1i/2)^2 + 1i/2.
- For c = 2, the second term is 2^2 + 2.
3. Repeat this process to find the subsequent terms until you have the desired number of terms.
Let's calculate the first six terms for both values of c:
For c = 1i/2:
- First term: 1i/2
- Second term: (1i/2)^2 + 1i/2 = -1/4i + 1i/2 = (1 - 2i)/4
- Third term: ((1 - 2i)/4)^2 + 1i/2 = (1 - 4i + 4i^2)/16 + 1i/2 = (-4 - 4i)/16 + 1i/2 = (-1 - i)/4
- Fourth term: ((-1 - i)/4)^2 + 1i/2 = (1 + 2i + i^2)/16 + 1i/2 = (2 + 1i)/16 + 1i/2 = (2 + 17i)/32
- Fifth term: ((2 + 17i)/32)^2 + 1i/2 = (4 + 68i + 289i^2)/1024 + 1i/2 = (293 + 44i)/1024
- Sixth term: ((293 + 44i)/1024)^2 + 1i/2 = (418225 + 64316i)/1048576
For c = 2:
- First term: 2
- Second term: 2^2 + 2 = 6
- Third term: 6^2 + 2 = 38
- Fourth term: 38^2 + 2 = 1446
- Fifth term: 1446^2 + 2 = 2090438
- Sixth term: 2090438^2 + 2 = 4368655165762
So, the first six terms for c = 1i/2 are: (1i/2), ((1 - 2i)/4), ((-1 - i)/4), ((2 + 17i)/32), ((293 + 44i)/1024), ((418225 + 64316i)/1048576).
And the first six terms for c = 2 are: (2), (6), (38), (1446), (2090438), (4368655165762).